True or false questions about countable and non-countable sets The set of all power series with integer coefficients is countable
The set of all polynomials with integer coefficients is countable
I am not sure about the first statement.. 
I think second statement is true if we consider polynomials one by one. Namely, first consider constant polynomial which is just all integers, then consider first degree polynomial, which is equal to Z x Z. However, I can't not think of any formal way of writing the proof.
 A: Your idea about second statement goes right way. To proceed your proof, you shall check that the set of all polynomials with degree $\le n$ and integer coefficient is equipotent with $\Bbb{Z}^{n+1}$, and use the fact that a countable union of countable sets is countable.
First statement is false. You can use diagonal argument, or you can construct an injection from $\mathcal{P}(\Bbb{N})$ to the set.
For construct an injection, you can consider following series, for each $A\subseteq \Bbb{N}$:
$$f_A(x) = \sum_{n\in A} x^n$$
and you can show that $A \mapsto f_A$ is one-to-one.
A: For the first one, note that the set of all power series whose coefficients are in the set $\{0,1,2,3,\dots,9\}$ is a subset of all power series whose coefficients are integers (since the former is more restrictive).
Now, note that these more restricted power series are in direct bijection with real numbers in the range of $[0,1]$.  Since any interval of real numbers is uncountable, and our subset of restricted power series is in bijection with it, it must also be uncountable.
Finally, since any set which has an uncountable subset is uncountable, the original set must be uncountable as well.
